ORIGINAL ARTICLE: Matteo Broso, Privatization under Political Ties. https://www.matteobroso.com/
This is still a working paper (not peer-reviewed).
This is still a working paper (not peer-reviewed).
Article Abstract: A politically tied public firm competes with a private one in a game of endogenous product differentiation. Consumers differ in their willingness to pay. Firms choose qualities and then prices. Finally, consumers buy one good. The public firm maximizes a combination of profits and the median voter’s utility. I provide a political economy micro-foundation of this objective function. Then, I study the effects of partial privatization. I find that (i) privatization encourages the entry of the private firm; (ii) the public firm’s profits may exceed those of its competitor; and (iii) the socially optimal degree of privatization is interior.
Introduction
Introduction
This notebook is attached to the Working Paper: Privatization under Political Ties (available on my website). Supplementary notebooks are attached below.
In this notebook, I solve for NE prices in the Subgame induced by a generic pair of qualities if λ>1/2. This corresponds to Lemma 1 in the paper. I will show that there exist three different class of NE. I fully characterize each class and its region of existence in the quality space.
In the other attached notebooks, I characterize the SPNE (Proposition 1). In particular:
- In CANDIDATE_SPNE_PL.nb, I obtain (candidate) SPNE qualities by solving the FOCs inside of one of the price NE regions.
- In LEAPFROGGING_DEVIATIONS.nb, I study if candidate SPNE qualities are subject to profitable unilateral leapfrogging deviations.
- In INTER_REGIONS_DEVIATIONS.nb, I check whether, given candidate SPNE qualities, firms want to reach another price region, and I show the existence of other four pair of SPNE quality pairs (if is high enough). I also recap the SPNE qualities and perform a welfare analysis.
In MARGINAL_COST_PRICING.nb I show the computations for Proposition 4.
It is worth noting that:
- In the paper, I discuss why deviations towards monopolistic pricing are not optimal. This is the reason why I can characterize the price NE (and the SPNE) only if . Everything in these notebooks applies if (even if not explicitly indicated)
- In the attached notebooks, I provide the proofs of SPNE characterization using a normalization in some parameters. As I discuss in the paper, this comes without loss of generality.
- These notebooks may have possibly have an inconsistent notation with the paper’s and among each other. I clarify the notation at the beginning of each notebook. For instance, firm 0 is firm S. Firm 1 is firm P.
- In CANDIDATE_SPNE_PL.nb, I obtain (candidate) SPNE qualities by solving the FOCs inside of one of the price NE regions.
- In LEAPFROGGING_DEVIATIONS.nb, I study if candidate SPNE qualities are subject to profitable unilateral leapfrogging deviations.
- In INTER_REGIONS_DEVIATIONS.nb, I check whether, given candidate SPNE qualities, firms want to reach another price region, and I show the existence of other four pair of SPNE quality pairs (if
λ
In MARGINAL_COST_PRICING.nb I show the computations for Proposition 4.
It is worth noting that:
- In the paper, I discuss why deviations towards monopolistic pricing are not optimal. This is the reason why I can characterize the price NE (and the SPNE) only if
λ>1/2
λ>1/2
- In the attached notebooks, I provide the proofs of SPNE characterization using a normalization in some parameters. As I discuss in the paper, this comes without loss of generality.
- These notebooks may have possibly have an inconsistent notation with the paper’s and among each other. I clarify the notation at the beginning of each notebook. For instance, firm 0 is firm S. Firm 1 is firm P.
Analysis
Analysis
Consumers are uniformly distributed along
In[]:=
Consumers={θ-1,θ}
Out[]=
{-1+θ,θ}
Profits of firm s are
In[]:=
πs=xs(ps-αqs^2)
Out[]=
xs(ps-α)
2
qs
Profits of firm p are
In[]:=
πp=xp(pp-αqp^2)
Out[]=
xp(pp-α)
2
qp
In the price stage, firms have observed qualities. FOCs are sufficient by concavity. I distinguish between two cases: qs<qp; qs>qp. The case qs=qp is trivial.
qs<qp
qs<qp
The indifferent consumer is
In[]:=
θisl=(pp-ps)/(qp-qs)
Out[]=
pp-ps
qp-qs
The demands are
In[]:=
xs=(θisl-(θ-1))
Out[]=
1+-θ
pp-ps
qp-qs
In[]:=
xp=(θ-θisl)
Out[]=
-+θ
pp-ps
qp-qs
(Duopoly) Profits are
In[]:=
πs
Out[]=
(ps-α)1+-θ
2
qs
pp-ps
qp-qs
In[]:=
πsl[ps_,pp_,qs_,qp_]:=(ps-α)1+-θ
2
qs
pp-ps
qp-qs
In[]:=
πp
Out[]=
(pp-α)-+θ
2
qp
pp-ps
qp-qs
In[]:=
πpsl[ps_,pp_,qs_,qp_]:=(pp-α)-+θ
2
qp
pp-ps
qp-qs
Consumer surplus writes as follows
In[]:=
cssl[ps_,pp_,qs_,qp_]:=Integrate[xqs-ps,{x,(θ-1),θisl}]+Integrate[xqp-pp,{x,θisl,(θ)}]
In[]:=
FullSimplify[cssl[ps,pp,qs,qp]]
Out[]=
2
pp
2
ps
2
qs
2qp-2qs
1
2
2
θ
Aggregate social welfare writes as follows
In[]:=
wsl[ps_,pp_,qs_,qp_]:=cssl[ps,pp,qs,qp]+πpsl[ps,pp,qs,qp]+πsl[ps,pp,qs,qp]
In[]:=
FullSimplify[wsl[ps,pp,qs,qp]]
Out[]=
-(++2ps(qp-qs)(qp+qs)α-2pp(ps+(qp-qs)(qp+qs)α)+(qp-qs)(-2α(-1+θ)+qs+qp(2qpα-θ)θ))
1
2(qp-qs)
2
pp
2
ps
2
qs
2
(-1+θ)
(Duopoly) Best Response Functions
(Duopoly) Best Response Functions
The median voter is
In[]:=
mv=((θ-1)+(θ))/2
Out[]=
1
2
Suppose that the median voter buys good s
In[]:=
mv<θisl
Out[]=
1
2
pp-ps
qp-qs
In[]:=
umed=(mv)qs-ps
Out[]=
-ps+qs(-1+2θ)
1
2
In[]:=
vs=(1-λ)umed+λπs
Out[]=
-ps+qs(-1+2θ)(1-λ)+(ps-α)1+-θλ
1
2
2
qs
pp-ps
qp-qs
In[]:=
FullSimplify[D[vs,ps]]
Out[]=
qs+(pp-2ps+qs(-2+qsα+θ))λ-qp(1+(-2+θ)λ)
qp-qs
In[]:=
FullSimplifyD,λ
qs+(pp-2ps+qs(-2+qsα+θ))λ-qp(1+(-2+θ)λ)
qp-qs
Out[]=
pp-2ps-qp(-2+θ)+qs(-2+qsα+θ)
qp-qs
In[]:=
FullSimplify[Solve[D[vs,ps]==0,ps]]
Out[]=
ps
qs+(pp+qs(-2+qsα+θ))λ-qp(1+(-2+θ)λ)
2λ
The optimal price of firm s is
In[]:=
psb1=
qs+(pp+qs(-2+qsα+θ))λ-qp(1+(-2+θ)λ)
2λ
Out[]=
qs+(pp+qs(-2+qsα+θ))λ-qp(1+(-2+θ)λ)
2λ
The optimal price of firm p is
In[]:=
FullSimplify[D[πp,pp]]
Out[]=
-2pp+ps+α
2
qp
qp-qs
In[]:=
FullSimplify[Solve[D[πp,pp]==0,pp]]
Out[]=
pp(ps+α+qpθ-qsθ)
1
2
2
qp
In[]:=
ppbest=(ps+α+qpθ-qsθ)
1
2
2
qp
Out[]=
1
2
2
qp
In[]:=
vs
Out[]=
-ps+qs(-1+2θ)(1-λ)+(ps-α)1+-θλ
1
2
2
qs
pp-ps
qp-qs
What is the maximum payoff that firm s can obtain in this case?
In[]:=
vsm=FullSimplify[Limit[vs,ps->psb1]]
Out[]=
1
4
qp-qs
λ
(-2pp(qs(2+qsα-θ)+qp(-2+θ))++2qpqs(-3+(2+qsα-θ)θ)+(2+))λ
2
pp
2
qp
2
(-2+θ)
2
qs
2
(-qsα+θ)
qp-qs
Suppose that the median voter buys good p (I repeat the same procedure above to find the maximum payoff that firm s can obtain by letting the median voter buy good p)
In[]:=
mv>θisl
Out[]=
1
2
pp-ps
qp-qs
In[]:=
umed1=(mv)qp-pp
Out[]=
-pp+qp(-1+2θ)
1
2
In[]:=
vs1=(1-λ)umed1+λπs
Out[]=
-pp+qp(-1+2θ)(1-λ)+(ps-α)1+-θλ
1
2
2
qs
pp-ps
qp-qs
In[]:=
FullSimplify[D[vs1,ps]]
Out[]=
(pp-2ps+qp-qpθ+qs(-1+qsα+θ))λ
qp-qs
In[]:=
FullSimplify[Solve[D[vs1,ps]==0,ps]]
Out[]=
ps(pp+qp-qpθ+qs(-1+qsα+θ))
1
2
I now show that firm s wants the median voter to buy good s if and only fi the price of good p is high enough
The following condition tells us when the median voter buys good s
Therefore we can write the best response of firm s as follows
The following cells show that both firms want to become monopolists if the price of their opponents is high enough
The best response of firm p is:
Pure Strategy NE
Pure Strategy NE
I now look for intersections of (duopoly) best responses
In the paper I argue that any of these intersections is safe from deviations towards monopoly
In the paper I argue that any of these intersections is safe from deviations towards monopoly
Equilibrium 1 (median voter buys good s)
Equilibrium 1 (median voter buys good s)
The conditions for the existence of this NE are
Let’s write down expected payoffs from this NE
Equilibrium 2 (median voter buys good 1)
Equilibrium 2 (median voter buys good 1)
NE Demand Functions are:
Expected Payoffs are as follows
Let me recap the condition of existence of the different NE (of course these cannot overlap)
The following shows that any time there exists this NE, xp>1/2 (median voter buys good p)
The following shows that these conditions cannot overlap
Mixed Strategy NE
Mixed Strategy NE
When it does not exist a pure strategy NE, I show that it exists a mixed strategy NE
Suppose that firm p plays the price such that firm s has two best responses, and firm s randomizes between these (of course this is optimal for firm s)
The two strategy in the support of firm s’s mixed strategy are:
The expected value of firm s’s mixed strategy is:
The expected value of firm p’s profits given firm s’s mixed strategy is
Solving for firm s’s mixed strategy NE
The mixed strategy NE requires that
Comparative Statics on Prices
Comparative Statics on Prices
Regions of Existence of the NE
Regions of Existence of the NE
The following cell shows the regions of existence of the different price NE
In the following cell, I show when firm 1 is out of the market in the different NE
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Privatization under Political Ties
by Matteo Broso
Wolfram Community, STAFF PICKS, September 21, 2023
https://community.wolfram.com/groups/-/m/t/3017901
by Matteo Broso
Wolfram Community, STAFF PICKS, September 21, 2023
https://community.wolfram.com/groups/-/m/t/3017901